The function e^x is the only possible function that is always equal to its own derivative. Its value (for any given x) is always the same as its slope (at that x). Any other function that is equal to its own derivative must be equivalent to e^x.

If you take the derivative of the infinite series

Σ (x^n)/n! = 1 + x/1! + x^2/2! + x^3/3! + …

each term in the series “shifts down” one. The 1 disappears, the x becomes a 1, the x^2/2! becomes x, the x^3/3! becomes x^2/2!, and so on. Because the series is infinite, this means that the derivative is of the series is equal to the series itself. Since it is its own derivative, it must be equal to e^x.

There’s a proof somewhere floating around. Since e is a constant and x is the variable, it is like taking a derivative of A to the x.

Some weird thing with natural logs, however, means that e to the x is its own derivative.

To me an even greater mystery is that e^x should be represented by

Σ (x^n)/n!

Maybe you could blame Taylor for that one.

I love the way the mathematicians define e: the limit as n goes to zero of the nth root of ( 1 + n). Identical to Jamie Irons‘s statement, but good for making a jaw drop. (Real mathematicians may be able to explain a difference, unknown to mere mortals, between the two formulations. The invitation is extended.)

1. Instapundit readers arrive at the site randomly, and once there they read all the posts they have not read yet. As time goes on more and more of the readers are repeat visitors, and they don’t re-read. To take one simple example, suppose there were 100 readers and each had a 50% chance of visiting Instapundit every hour (sounds about right!). In the first hour all 50 visitors would never have seen the link, and all would click. In the next hour, on average only 25 of the 50 would be new to the link. In the next hour, 12.5, and so on…

2. Instapundit readers tend to get distracted as they read down the page. Suppose 100 new readers arrive at the site every hour. They all read the top link, but only 95% of them make it down to the second link. Of those remaining, 95% make it to the third link. Now suppose that a link moves a fixed amount down the page every hour—say 10 slots. Because 0.95^10=0.6 this would mean that in the first hour you’d get 100 hits, in the second hour 60 hits, in the third hour 36…

About the time zones, you’ll notice that this isn’t really exponential decay because the first couple hours don’t decay fast enough. This may be the result of the west coast waking up—essentially, the arrival rate of new visitors would be increasing in the first few hours, counteracting the decay. Also, the 22nd hour might appear to be an outlier if it hadn’t been completed yet when the photo was taken.